The first attempt to classify all the multi-affine partial difference equations defined on the quad graph and possessing CAC was carried out
in [1]}. There the equation on the quad graph was treated as a geometric object not embedded in any \(\mathbb {Z}^{2}\) -lattice, as displayed in Fig. REF . The quad-equation is an expression of the form
\(Q( x,x_{1},x_{2},x_{12};\alpha _{1},\alpha _{2}) =0,\) 
